File:FS C(2C)C dia.png
![File:FS C(2C)C dia.png](https://upload.wikimedia.org/wikipedia/commons/thumb/9/98/FS_C%282C%29C_dia.png/600px-FS_C%282C%29C_dia.png?20231202161137)
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Captions
Captions
Summary
[edit]DescriptionFS C(2C)C dia.png |
English: Largest circle inscribed in a circle that contains a pair of equally sized circles
Deutsch: Größter Kreis, der in einen Kreis eingeschrieben ist, der zwei gleich große Kreise enthält |
Date | |
Source | Own work |
Author | Hans G. Oberlack |
![](https://upload.wikimedia.org/wikipedia/commons/thumb/9/98/FS_C%282C%29C_dia.png/220px-FS_C%282C%29C_dia.png)
Task
[edit]The largest circle inscribed in a circle of radius that contains two equally sized circles
General case
[edit]Segments in the general case
[edit]0) The radius of the base circle:
1+2) The radii of the inscribed twin circles:
3) The radius of the third inscribed circle: see calculation 1
Perimeters in the general case
[edit]0) Perimeter of base circle:
1+2) Perimeter of the inscribed twin circles:
3) Perimeter of third circle:
S) Sum of perimeters:
Areas in the general case
[edit]0) Area of the base circle:
1+2) Area of the inscribed twin circles:
3) Area of the third circle:
Centroids in the general case
[edit]1) Centroids as graphically displayed
[edit]0) Centroid position of the base circle:
1) Centroid position of the first inscribed circle measured from the centroid of the base shape:
2) Centroid position of the second inscribed circle measured from the centroid of the base shape:
3) Centroid position of the third inscribed circle measured from the centroid of the base shape:
2) Orientated centroids
[edit]The centroid positions of the following shapes will be expressed orientated so that the first shape n with will be of type
with
. This means that the graphical representation will not correspond to the ma-thematical expression.
0) Orientated centroid position of the base circle:
1) Orientated centroid position of the first inscribed circle measured from the centroid of the base shape:
2) Orientated centroid position of the second inscribed circle measured from the centroid of the base shape:
3) Orientated centroid position of the third inscribed circle measured from the centroid of the base shape:
Normalised case
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/6/68/FS_C%282C%29C.png/220px-FS_C%282C%29C.png)
In the normalised case the area of the base circle is set to 1.
So
Segments in the normalised case
[edit]0) Radius of the base circle
1+2) Radii of the inscribed circles:
3) Radius of the third circle:
Perimeters in the normalised case
[edit]0) Perimeter of base circle:
1+2) Perimeter of the inscribed circles:
3) Perimeter of third circle:
S) Sum of perimeters:
Areas in the normalised case
[edit]0) Area of the base square is by definition
1+2) Area of the inscribed circles:
3) Area of the third circle:
Covered surface of the base shape:
Centroids in the normalised case
[edit]1) Centroids as graphically displayed
[edit]0) Centroid position of the base circle:
1) Centroid position of the first inscribed circle measured from the centroid of the base shape:
2) Centroid position of the second inscribed circle measured from the centroid of the base shape:
3) Centroid position of the third inscribed circle measured from the centroid of the base shape:
2) Orientated centroids
[edit]The centroid positions of the following shapes will be expressed orientated so that the first shape n with will be of type
with
. This means that the graphical representation will not correspond to the ma-thematical expression.
0) Orientated centroid position of the base circle:
1) Orientated centroid position of the first inscribed circle measured from the centroid of the base shape:
2) Orientated centroid position of the second inscribed circle measured from the centroid of the base shape:
3) Orientated centroid position of the third inscribed circle measured from the centroid of the base shape:
Calculations
[edit]Given elements
[edit](1)
(2)
(3) because the circles are tangent to each other
(4)
Calculation 1
[edit], applying the Pythagorean theorem on right triangle
, applying equation (2)
, applying equation (4)
, applying equation (3)
, applying binomial formulas
, rearranging
, rearranging
, applying equation (2)
, rearranging
, rearranging
, rearranging
, rearranging
Calculation 2
[edit]
, since
, since
, since
, shortening
, shortening
, since
, rearranging
, shortening
, since
, rearranging
Licensing
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File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 16:11, 2 December 2023 | ![]() | 1,305 × 1,305 (108 KB) | Hans G. Oberlack (talk | contribs) | Diagram refined |
10:56, 27 May 2023 | ![]() | 1,305 × 1,305 (109 KB) | Hans G. Oberlack (talk | contribs) | Uploaded own work with UploadWizard |
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